Activity Dynamics and Signal Representation in a Striatal Network Model with Distance-Dependent Connectivity

Spiking activity of the network with gamma-distributed (top) and Gaussian-distributed (bottom) connectivity. (a) Connection probability between neurons as a function of the distance between neurons normalized to the full size of the network. (b) Examples of the spiking activity of 200 of 10,000 neurons for different average firing rates ( ) of the network. With increasing background activity, the spiking activity of the Gaussian networks remained irregular and the bursting behavior increased. In contrast, the gamma networks showed transient or persistent bursting behavior and local synchronization of spiking activities.

Analysis of the spike patterns in networks with different connectivity profiles. a, Distribution of average firing rates ( ; top) and coefficient of variation (CV_ISI; bottom) as a function of background input strength (ν_ext) for Gaussian (left) and gamma (right) networks. In Gaussian networks, increasing ν_ext resulted in a steadily widening distribution of and CV_ISI, which for a large fraction of neurons tended to the value 3. In contrast, in gamma networks, the distribution of and CV_ISI was rapidly widely distributed from a relatively low ν_ext (1.5 kHz) upward. Gamma networks were clearly more excitable than Gaussian networks. Green trace indicates the skewness of the firing rate in both networks. Black and blue traces refer to the average firing rate ( ) and standard deviation (σ_λ) of the firing rate distribution, respectively. b, Relationship between the irregularity of the spiking pattern (CV_ISI) and the average firing rate ( ). The color of the traces represents the background input rate (ν_ext). In Gaussian networks (top), neurons with higher tended to exhibit a higher CV_ISI, whereas in gamma networks (bottom), they tended to exhibit a lower CV_ISI.

Characterization of the bump activity states for the gamma network. Time series snapshots of the 2D pattern (100 × 100 neurons) of bump activity, contrast-enhanced by Mexican hat filtering. Each frame was measured by summing the neuronal activity over 100 ms. Three snapshots (columns) of activity were taken 1 s apart. Three representative gamma network states for three different amounts of external inputs are shown in each row. AI: external input = 1 kHz; no bump activity is observed and the network activity remains noisy. TA: external input = 1.5 kHz; the network is in an unstable state, with several bumps appearing transiently, in the company of noisy activity. WTA: external input = 3 kHz; the network forms mostly persistent bumps throughout the entire network.

Quantification of bump activities in gamma networks. a, Spatial autocorrelation of the network spiking activity, showing the mean firing rate ( ) of each bump as a function of the distance from the cluster centroid normalized to the full size of the network. Different colors represent the strength of the background excitation. b, Distance between bumps in various gamma distributions and its comparison between numerical simulations and mean field equations. The background color is measured bump distance from the network simulation data. Solid traces show results from analytical estimation of bump distance; dashed traces represent the estimation of bump distance from network simulations. The parameter for the gamma-distributed connection is used for the spiking network model (red circle). Spatial representations of activity bumps are also observed in Fig. 5 (c–e). These subfigures display the bump count and their relative lifespan during the entire simulation as a function of the background external excitation (ν_ext), which modulates the nature of the bump activity. c, With increasing ν_ext, the number of bumps increases in a sigmoidal fashion. For higher ν_ext, the number of bumps saturates, owing to the limited capacity of the finite spatial map. The error bars for the bump counts indicate the standard deviation of bump counts over the simulation time. d, The lifespan of bumps reflects the dynamic state of the bumps: a shorter lifespan reflects TA dynamics, whereas a larger lifespan indicates stable bump activity reflecting WTA dynamics. By increasing ν_ext, the distribution of lifespans shifts from short to long term. The ordinate indicates lifespan, normalized to the duration of the entire simulation (10 s), of individual bumps. Because the average bump counts are different in each dynamical state (TA, WTA), we normalized the color bar of bump count to the average bump count in individual states. e, The lifespan distribution is split into three groups (dashed lines in d). The long (red trace) appearance of bumps reflects the WTA state of bump activity, whereas the short (blue trace) appearance of bumps reflects the TA state of bump activity. Between these two states, the network is in a highly unstable state, characterized by a wider distribution of lifespans (light blue trace).

Spatial map of bump activity patterns for different gamma distributions. A snapshot, contrast-enhanced by Mexican hat filtering, of the 2D pattern (100 × 100 neurons) of bump activity for different parameters (shape, scale) of gamma connection profile defines the size of bumps and the distance between bumps. The rate of the external Poisson excitation (ν_ext) was set to 5 kHz to obtain WTA states in networks with different gamma distributions.

Analytical description of connectivity profiles. The graphs show the spatial connectivity profile (left) and its Fourier transform (right) as a function of the distance between neurons and wave numbers, respectively, for both gamma (red) and Gaussian (blue) connectivity kernels. Note that the spatial connectivity profile remains positive for both connectivity kernels. However, their Fourier transforms behave differently: the Gaussian kernel remains positive, whereas the gamma kernel takes negative values for larger (absolute) wave numbers.

Impact of the network dynamics on the stimulus response. a, Spatial distribution of the spiking activity displayed in time series for different dynamic states (rows). Each frame shows a spatial map of 30 × 30 neurons from the ROI (black squares) in a time window of 100 ms, with 500-ms intervals between successive frames. The gradient background is the probability area for stimulated neurons, and its color refers to a stimulus phase. b, The change of response activity of the stimulated neurons to their corresponding stimuli A (red) or B (orange) in different ongoing bump states (AI, TA, and WTA). Each row represents the strength of the stimuli (50–150 pA). A lower Δresponse indicates a weaker impact of the external stimuli on the network activity, and lower variance of activity reflects a higher reliability of the response. For each subpanel, the white lines are the median value of the data. The colored boxes extend from the 25% to 75% of the data, i.e., the box contains ≈50% of the data. Whiskers extend from minimum to maximum values of the data. c, The temporal variability (FF) of the response of the stimulated neurons as a function of time. A lower FF indicates a higher reliability of the stimulus response. A higher FF is observed at each stimulus onset, in both the AI and TA states. In contrast, in the WTA state, the network is not able to reliably respond to external stimuli. Both stimulus phases are displayed at the bottom of each subpanel.